trigonometric sum identity

I'm looking for a hint on how to prove that

I know I'm on here a lot but I really have no one to go to for guidance and am trying to learn this material on my own from the text book. Thank you for being patient with me so far and for being so helpful.

I know I'm on here a lot but I really have no one to go to for guidance and am trying to learn this material on my own from the text book. Thank you for being patient with me so far and for being so helpful.

You might consider using the formula for the sum of a geometric series to find and then getting the real part of the result.

Thanks guys. I was able to do it via induction but I think it was intended that I use some more direct method. It says "obtain the formula" I tried complexification as mr. fantastic recommended and obtained some interesting results but nothing I could use trig identities to simplify into the required form.

Also I was wondering if there is a closed form of or . I figured if there is such a formula it might come in handy here.

I've also tried using the infinite series definition of cosine and sine and again came up with interesting results but nothing that would help me unless I had a formula for the series I just mentioned.

Thank you two for your help thus far. I really just wish my book had more examples that I could build on for these problems. Does anyone know of some catalogue of proofs that I could look at to get some more inspiration?

Thanks guys. I was able to do it via induction but I think it was intended that I use some more direct method. It says "obtain the formula" I tried complexification as mr. fantastic recommended and obtained some interesting results but nothing I could use trig identities to simplify into the required form.

[snip]

You should show your working.

I get and the real part of this is . Your job is to get these two expressions.

Originally Posted by magus

[snip]

Also I was wondering if there is a closed form of or .

[snip]

Yes there are. But they are not relevant to the proof you are working on.

Last edited by mr fantastic; September 29th 2010 at 04:57 AM.
Reason: Merged posts, fixed small typo (had k instead of n in results), and a bigger typo.

Is this right so far? Because what comes after when I apply Euler's identity again is a god forsaken mess when I try to get the real part.

Just to show I'm doing the work though

Then comes the process of making the denominator real by multiplying by a conjugate over a conjugate. I'd show you the work I've done for that but unfortunately the size limit for the LaTeX images created prohibits me from doing so.

Is this right so far? Because what comes after when I apply Euler's identity again is a god forsaken mess when I try to get the real part.

Just to show I'm doing the work though

Then comes the process of making the denominator real by multiplying by a conjugate over a conjugate. I'd show you the work I've done for that but unfortunately the size limit for the LaTeX images created prohibits me from doing so.

How does this reduce?

(think carefully because this is what you essentially have) and I have told you what a and r are.

After substitution you get .

Getting the real part of this expression should be trivial at this level (but making careless mistakes will make it seem a lot harder I suppose).